joint estimation of multiple regression coefficient matrices
Description
wfrl
estimates jointly two regression coefficient matrices from multivariate
normal distributed datasets using an ADMM based algorithm.
Usage
1 2 3 
Arguments
D1 
list with the response variables. Two matrices in the list corresponding to the response variables of the two populations. 
D2 
list with the explanatory variables. Two matrices in the list corresponding to the explanatory variables of the two populations. 
lambda1 
tuning parameter for sparsity in the regression coefficients. 
lambda2 
tuning parameter for similarity between the regression coefficients in the two populations. 
automLambdas 
if 
paired 
if 
sigmaEstimate 
robust method used to estimate the variance of estimated partial correlations: name that uniquely identifies

maxiter 
maximum number of iterations for the ADMM algorithm. 
tol 
convergence tolerance. 
nsubset 
maximum number of estimated partial correlation coefficients (chosen randomly) used to select 
rho 
regularization parameter used to compute matrix inverse by eigen value decomposition (default of 1). 
rho.increment 
default of 1. 
notOnlyLambda2 
if 
Details
wfrl
uses a weightedfused least squares lasso maximum likelihood estimator by solving:
[\hat{β}_H,\hat{β}_T] = \arg\min\limits_{β_H,β_T} ≤ft[ \frac{1}{2n}Yβ_HX^2_2 + \frac{1}{2n}Qβ_TW^2_2 +P_{λ_1,λ_2,V}(β)\right]
with
P_{λ_1,λ_2,V}(β) = λ_1β_H_1 + λ_1β_T_1 + λ_2V \circ (β_Tβ_H)_1.
where λ_1 is the sparsity tuning parameter, λ_2 is the similarity tuning parameter, and V = [v_{ij}]
is a p\times p matrix to weight λ_2 for each coefficient of the differential precision matrix.
If datasets are independent (paired = "FALSE"
), then it is assumed that v_{ij} = 1 for all pairs (i,j).
Otherwise (paired = "TRUE"
), weights are estimated in order to account for the dependence structure between datasets in the differential
network estimation. An ADMMtype recursive algorithm is used to solve the optimization problem.
See details in wfgl
for transforming the selection problem of the tuning parameters λ_1 and λ_2.
Value
An object of class wfrl
containing the following components:
regCoef 
regression coefficients. 
path 
nonzero structure for the regression coefficients. 
diff_value 
convergence control. 
iters 
number of iterations used. 
Author(s)
Caballe, Adria <a.caballe@sms.ed.ac.uk>, Natalia Bochkina and Claus Mayer.
References
Danaher, P., P. Wang, and D. Witten (2014). The joint graphical lasso for inverse covariance estimation across multiple classes. Journal of the Royal Statistical Society: Series B (Statistical Methodology) (2006), 120.
Boyd, S. (2010). Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. Foundations and Trends in Machine Learning 3(1), 1122.
See Also
plot.wfrl
for graphical representation.
wfgl
for joint partial correlation estimation.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 
# example to use of wfrl
N < 200
EX2 < pcorSimulatorJoint(nobs = N, nclusters = 3, nnodesxcluster = c(30, 30,30),
pattern = "pow", diffType = "cluster", dataDepend = "diag",
low.strength = 0.5, sup.strength = 0.9, pdiff = 0.5, nhubs = 5,
degree.hubs = 20, nOtherEdges = 30, alpha = 2.3, plus = 0,
prob = 0.05, perturb.clust = 0.2, mu = 0, diagCCtype = "dicot",
diagNZ.strength = 0.6, mixProb = 0.5, probSign = 0.7,
exactZeroTh = 0.05)
P < EX2$P
q < 50
BETA1 < array(0, dim = c(P, q))
diag(BETA1) < rep(0.35,q)
BETA2 < BETA1
diag(BETA2)[c(1:floor(q/2))] < 0
sigma2 < 1.3
Q < scale(EX2$D1)
W < scale(EX2$D2)
X < Q%*%BETA1 + mvrnorm(N,rep(0,q),diag(rep(sigma2,q)))
Y < W%*%BETA2 + mvrnorm(N,rep(0,q),diag(rep(sigma2,q)))
D1 < list(scale(X), scale(Y))
D2 < list(scale(Q), scale(W))
## not run
#wfrl1 < wfrl(D1, D2, lambda1 = 0.05, lambda2 = 0.05, automLambdas = TRUE, paired = FALSE,
# sigmaEstimate = "CRmad", maxiter = 30, tol = 1e05, nsubset = 10000, rho = 1,
# rho.increment = 1, notOnlyLambda2 = TRUE)
#print(wfrl1)
